integral from 1/3 to 5 of 12xln(3x)

解

\int_{\frac{1}{3}}^{5} 12 x ln (3 x) d x

12 (- \frac{56}{9} + \frac{25}{2} ln (15))

十進法表記

331.54086 \dots

解答ステップ

\int_{\frac{1}{3}}^{5} 12 x ln (3 x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 12 \cdot \int_{\frac{1}{3}}^{5} x ln (3 x) d x

部分積分を適用する

= 12 [\frac{1}{2} x^{2} ln (3 x) - \int \frac{x}{2} d x]_{\frac{1}{3}}^{5}

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= 12 [\frac{1}{2} x^{2} ln (3 x) - \frac{x ^{2}}{4}]_{\frac{1}{3}}^{5}

限度を計算する:

- \frac{56}{9} + \frac{25}{2} ln (15)

= 12 (- \frac{56}{9} + \frac{25}{2} ln (15))